NCERT Class 12-Mathematics: Chapter –1 Relations and Functions Part 6
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Question 4:
Let be the function defined by . write .
Answer:
Question 5:
If and the function , write .
Answer:
Question 6:
If is defined by , write .
Answer:
Question 7:
Is a function? If is described by , then what value should be assigned to and .
Answer:
Question 8:
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
(i) .
(ii).
Answer:
(i) Represents function which is surjective but not injective
(ii) Does not represent function.
Question 9:
If the mappings f and g are given by
and , write .
Answer:
Question 10:
Let C be the set of complex numbers. Prove that the mapping given by , is neither one-one nor onto.
Answer:
We have,
given by
In order to prove that f is one-one, it is sufficient to prove that, .
Let and are two distinct complex numbers.
Now,
Here, we observe that
This shows that different element of C may have the same value in R.
Thus, is not one-one.
is onto if every element of R is the f-image of some element of C.
We have, and
We observe that negative real numbers in R do not have their pre-images in C.
Thus, is not onto.
Hence, is neither one-one nor onto.
Question 11:
Let the function be defined by . Show that is neither one-one nor onto.
Answer:
We have,
In order to prove that f is one-one, it is sufficient to prove that , .
Let and are two different elements in R.
Now,
We observe that .
This shows that different element in R may have same image.
Thus, is not one-one.
We know that lies between and .
So, the range of f is which is not equal to its co-domain.
i.e., range of
In other words, range of is less than co-domain, i.e. there are elements in co-domain which does not have any pre-image in domain.
So, f is not onto.
Hence, is neither one-one nor onto.